On radial Schroedinger operators with a Coulomb potential

TL;DR

This paper analyzes one-dimensional Schrödinger operators with Coulomb and centrifugal potentials, introducing a complex-parameter family called Whittaker operators, and studies their spectral and scattering properties relevant to quantum mechanics.

Contribution

It introduces a two-parameter holomorphic family of Whittaker operators with complex coupling constants and analyzes their spectral and scattering theory.

Findings

Spectral properties of Whittaker operators are characterized.

Scattering theory for these operators is developed.

Operators generalize radial Schrödinger operators with Coulomb potential.

Abstract

This paper presents a thorough analysis of 1-dimensional Schroedinger operators whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. We allow both coupling constants to be complex. Using natural boundary conditions at 0, a two parameter holomorphic family of closed operators is introduced. We call them the Whittaker operators, since in the mathematical literature their eigenvalue equation is called the Whittaker equation. Spectral and scattering theory for Whittaker operators is studied. Whittaker operators appear in quantum mechanics as the radial part of the Schroedinger operator with a Coulomb potential.